nach oben

Designs, Codes and Cryptography

Erschienen in:

01.06.2015

Fourier-reflexive partitions and MacWilliams identities for additive codes

verfasst von: Heide Gluesing-Luerssen

Erschienen in: Designs, Codes and Cryptography | Ausgabe 3/2015

Einloggen, um Zugang zu erhalten

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

A partition of a finite abelian group gives rise to a dual partition on the character group via the Fourier transform. Properties of this dualization are investigated, and a convenient test is given for when the bidual partition coincides with the primal partition. Such partitions permit MacWilliams identities for the partition enumerators of additive codes. It is shown that dualization commutes with taking products and symmetrized products of partitions on cartesian powers of the given group. After translating the results to Frobenius rings, which are identified with their character module, the approach is applied to partitions that arise from poset structures

Vorheriger Artikel False positive probabilities in q-ary Tardos codes: comparison of attacks

Nächster Artikel Worst-case to average-case reductions for module lattices

The author is grateful to the anonymous reviewer for suggesting this line of proof.

In [9, 43], the authors use a reversed inner dot product. This results in the partition \({\mathcal P}_{{\mathbf P}}\) being self-dual, and no dual poset is needed.

Barra A., Gluesing-Luerssen H.: MacWilliams extension theorems and the local-global property for codes over Frobenius rings. Preprint, submitted. arXiv:1307.7159v1 (2013).

Brualdi R.A., Graves J., Lawrence K.M.: Codes with a poset metric. Discret. Math. 147, 57–72 (1995).

Byrne E., Greferath M., O’Sullivan M.E.: The linear programming bound for codes over finite Frobenius rings. Des. Codes Cryptogr. 42, 289–301 (2007).

Camion P.: Codes and association schemes. In: Pless V.S., Huffman W.C. (eds.) Handbook of Coding Theory, vol. II, pp. 1441–1566. Elsevier, Amsterdam (1998).

Claasen H.L., Goldbach R.W.: A field-like property of finite rings. Indag. Math. 3, 11–26 (1992).

Delsarte P.: An algebraic approach to the association schemes of coding theory. Philips Res. Rep. Suppl. 10, 1–97 (1973).

Delsarte P.: Bilinear forms over a finite field, with applications to coding theory. J. Comb. Theory Ser. A 25, 226–241 (1978).

Delsarte P., Levenshtein V.: Association schemes and coding theory. IEEE Trans. Inf. Theory 44, 2477–2504 (1998).

Dougherty S., Skriganov M.: MacWilliams duality and the Rosenbloom–Tsfasman metric. Moscow Math. J. 2(1), 81–97 (2002).

10.

El-Khamy M., McEliece R.J.: The partition weight enumerator of MDS codes and its applications. In: Proceedings of the IEEE International Symposium on Information Theory (ISIT 2005), Adelaide, Australia, pp. 926–930 (2005).

11.

Forney Jr., G.D.: Transforms and groups. In: Vardy A. (ed.) Codes, Curves and Signals: Common Threads in Communications, pp. 79–97. Kluwer, Boston (1998).

12.

Ganesan A., Vontobel P.: Vontobel. On the existence of universally decodable matrices. IEEE Trans. Inf. Theory 53, 2572–2575 (2007).

13.

Gluesing-Luerssen H.: Partitions of Frobenius rings induced by the homogeneous weight. Preprint, accepted for publication in Adv. Math. Commun. arXiv:1304.6589v1 (2013).

14.

Gluesing-Luerssen H.: On MacWilliams identities for codes over rings. In: Hüper, K., Trumpf, J. (eds.) Mathematical System Theory—Festschrift in Honor of Uwe Helmke on the Occasion of his Sixtieth Birthday, pp. 167–182. CreateSpace, Scotts Valley (2013).

15.

Godsil C.D.: Algebraic Combinatorics. Chapman and Hall, New York (1993).

16.

Hammons A.R., Kumar P.V., Calderbank A.R., Sloane N.J.A., Solé P.: The \({\mathbb{z}}_4\)-linearity of Kerdock, Preparata, Goethals, and related codes. IEEE Trans. Inf. Theory 40, 301–319 (1994).

17.

Hirano Y.: On admissible rings. Indag. Math. 8, 55–59 (1997).

18.

Honold T.: Characterization of finite Frobenius rings. Arch. Math. 76, 406–415 (2001).

19.

Honold T.: Two-Intersection sets in projective Hjelmslev spaces. In: Proceedings of the 19th International Symposium on the Mathematical Theory of Networks and Systems, pp. 1807–1813, Budapest, Hungary (2010).

20.

Honold T., Landjev I.: Linear codes over finite chain rings. Electron. J. Comb. 7 (2000).

21.

Honold T., Landjev I.: MacWilliams identities for linear codes over finite Frobenius rings. In: Jungnickel D., Niederreiter H. (eds.) Proceedings of the Fifth International Conference on Finite Fields and Applications Fq5 (Augsburg 1999), pp. 276–292. Springer, Berlin (2001).

22.

Huffman C.: On the theory of \({\mathbb{F}}_q\)-linear \({\mathbb{F}}_{q^t}\)-codes. Adv. Math. Commun. 7, 349–378 (2013).

23.

Huffman W.C., Pless V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003).

24.

Kim D.: Dual MacWilliams pair. IEEE Trans. Inf. Theory 51, 2901–2905 (2005).

25.

Kim H., Oh D.: A classification of posets admitting the MacWilliams identity. IEEE Trans. Inf. Theory 51, 1424—1431 (2005).

26.

Klemm M.: Über die Identität von MacWilliams für die Gewichtsfunktion von Codes. Arch. Math. (Basel) 49, 400–406 (1987).

27.

Klemm M.: Selbstduale Codes über dem Ring der ganzen Zahlen modulo \(4\). Arch. Math . (Basel) 53, 201–207 (1989).

28.

Lam T.Y.: Lectures on Modules and Rings. Graduate Text in Mathematics, vol. 189. Springer, Berlin (1999).

29.

Lamprecht E.: Über I-reguläre Ringe, reguläre Ideale and Erklärungsmoduln. I. Math. Nachr. 10, 353–382 (1953).

30.

Landjev I.: A note on the MacWilliams identities. Compt. Rend. Bulg. Acad. Sci. 52 (1999).

31.

Lu H., Kumar P.V., Yang E.: On the input-output weight enumerators of product accumulate codes. IEEE Commun. Lett. 8, 520–522 (2004).

32.

MacWilliams F.J.: Combinatorial problems of elementary abelian groups. Ph.D. Thesis, Harvard University (1962).

33.

MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1977).

34.

Nechaev A.A.: Linear codes over modules and over spaces: MacWilliams identities. In: Proceedings of the 1996 IEEE International Symposium on Information Theory and Application, Victoria, BC, Canada, pp. 35–38 (1999).

35.

Nechaev A.A., Kuzmin A.S.: Formal duality of linearly presentable codes over a Galois field. In: Mora T., Mattson H. (eds.) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. Lecture Notes in Computer Science, vol. 1255, pp. 263–276. Springer, Berlin (1997).

36.

Niederreiter H.: A combinatorial problem for vector spaces over finite fields. Discret. Math. 96, 221–228 (1991).

37.

Park W., Barg A.: Linear ordered codes, shape enumerators and parallel channels. In: 48th Annual Allerton Conference on Communication, Control, and Computing, Allerton, pp. 361–367 (2010).

38.

Pinheiro J., Firer M.: Classification of poset-block spaces admitting MacWilliams type identity. IEEE Trans. Inf. Theory 58, 7246–7252 (2012).

39.

Rosenbloom M., Tsfasman M.: Codes for the \(m\)-metric. Problemy Peredachi Informatsii 33(1), 55–63 (1997).

40.

Simonis J.: MacWilliams identities and coordinate positions. Lin. Algebra Appl. 216, 81–91 (1995).

41.

Stanley R.P.: Enumerative Combinatorics, vol. I. Cambridge University Press, Cambridge (1997).

42.

Terras A.: Fourier Analysis on Finite Groups and Applications. London Mathematical Society Student Texts, vol. 43. Cambridge University Press, Cambridge (1999).

43.

Trinker H.: A simple derivation of the MacWilliams identity for linear ordered codes and orthogonal arrays. Des. Codes Cryptogr. 50, 229–234 (2009).

44.

van Lint J.H. Introduction to Coding Theory, 3rd edn. Springer, Berlin (1999).

45.

Wood J.A.: Duality for modules over finite rings and applications to coding theory. Am. J. Math. 121, 555–575 (1999).

46.

Yang S.: Second-order weight distributions. IEEE Trans. Inf. Theory 57, 6068–6077 (2011).

47.

Zinoviev V.A., Ericson T.: On Fourier invariant partitions of finite abelian groups and the MacWilliams identity for group codes. Probl. Inf. Transm. 32, 117–122 (1996).

48.

Zinoviev V.A., Ericson T.: Fourier invariant pairs of partitions of finite abelian groups and association schemes. Probl. Inf. Transm. 45, 221–231 (2009).

Titel: Fourier-reflexive partitions and MacWilliams identities for additive codes
verfasst von: Heide Gluesing-Luerssen
Publikationsdatum: 01.06.2015
Verlag: Springer US
Erschienen in: Designs, Codes and Cryptography / Ausgabe 3/2015
Print ISSN: 0925-1022
Elektronische ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-014-9940-x

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Weitere Artikel der Ausgabe 3/2015

Hanani triple packings and optimal -ary codes of constant weight three

New results on nonexistence of generalized bent functions

New cube root algorithm based on the third order linear recurrence relations in finite fields

Permutation codes invariant under isometries

Worst-case to average-case reductions for module lattices

Difference matrices related to Sophie Germain primes using functions on the fields